You're right Bill, I can't prove it doesn't exist, but the idea that I need to do so is your logical fault, not my own.
And your logical error is the classic conspiracy theorist tactic of operating in the sceptical context. You do this with all the evidence which indicates that atomic weapons are a thing, that they've been used twice in anger in living memory and that there have been loads of atomic tests since, all of which have radioactive signatures you wouldn't find in conventional bombs. Then there's the fact that the Hiroshima bomb was the equivalent of 16,000 tons of TNT, not quite sure how the Enola Gay could have transported that. None of the evidence is good enough for you. You dismiss it all.
But you do this selectively. The stuff you want to believe - that Hiroshima was "just firebombing", you claim things like that without providing any evidence. And then you do a load of wild speculation that if nuclear weapons were a thing then <bad things> would happen. But that is just you speculating, it's not evidence of anything.
So no, you can't prove nuclear weapons don't exist any more than anyone can prove they do - not to the standard you demand. But some evidence for your claims would be nice.
You can use this tactic to believe - or disbelieve - anything you like.
"Kangaroos don't exist."
"Here's a photo of one."
"That's fake."
"OK, here's a video of one"
"CGI"
"I've been to Australia and seen some"
"You're mistaken or lying"
"OK, now we're at the zoo. Look, there's a kangaroo".
"Pah, that's just an animatronic fake".
And so on. You can do this about anything. If you have any good evidence for any of your claims then present it, otherwise it's just you making wild claims to back up your narrative.